⇒ Mathematical Logic

Correct answer option is (A)

Let $\mathrm{p}$ : The switch ${\mathrm{S}}_1$

$\mathrm{q}$ : The switch ${\mathrm{S}}_2$

The symbolic form is

$(p\wedge \sim q)\vee (\sim p\wedge q)\vee (\sim p\wedge \sim q)$

$\equiv (p\wedge \sim q)\vee [\sim p\wedge (q\vee \sim q)]$ ...[Distributive Law]

$\equiv (p\wedge \sim q)\vee [\sim p\wedge t]$ ...[Complement Law]

$\begin{array}{rl}& \equiv (p\wedge \sim q)\vee \sim p\text{...[Identity Law]}\\ & \equiv \sim p\vee (p\wedge \sim q)\text{... [Commutative Law]}\\ & \equiv (\sim p\vee p)\wedge (\sim p\vee \sim q)\text{...[Distributive Law]}\end{array}$

$\equiv t\wedge (\sim p\vee \sim q)$ ...[Complement Law]

$\equiv \sim p\vee \sim q$ ...[Identity Law]

$\therefore$ Option (A) is correct.

Correct answer option is (D)

The symbolic form of the given circuit is

$\begin{array}{rl}& (\mathrm{p}\vee \sim \mathrm{q}\vee \sim \mathrm{r})\wedge (\mathrm{p}\vee (\mathrm{q}\wedge \mathrm{r}))\\ & \equiv \mathrm{p}\vee [(\sim \mathrm{q}\vee \sim \mathrm{r})\wedge (\mathrm{q}\wedge \mathrm{r})]\text{...[Distributive law] }\end{array}$

$\equiv \mathrm{p}\vee [\sim (\mathrm{q}\wedge \mathrm{r})\wedge (\mathrm{q}\wedge \mathrm{r})]$ ...[De Morgan's law]

$\equiv \mathrm{p}\vee \mathrm{F}$ ....[Complement law]

$\equiv \mathrm{p}$ ...[Identity law]